When: after lights out
Without any particular prompting, this conversation started, probably as an excuse to stay up a couple more minutes.
J2: 23 isn't prime because it is divisible by .1
J0: Interesting. What number multiplied by .1 is 23?
J0: Hmm. 13 x 0.1 is thirteen 0.1s, so I think that would be 1.3
<J1 and J2 proceed to guess what values for 23/0.1>
J2: it has to be a multiple of 10
<a couple more guesses
J0: So, if you let yourself have fractions or decimals, then 23 isn't a prime. Another way to say that is 23 isn't a prime in the rationals.
J2 (initially copying): 23 isn't a prime in the rationals. Hey, nothing is prime in the rationals!
J0: Usually when we say "prime" by itself, we are just talking about the integers, like -2, -1, 0, 1, 2. if we stick to that, what do you think about 23?
J1 and J2 together: its prime!
J1: what about e, is it prime?
<brief continuation where I say that I know some other primes and extensions of the definition of prime, but I don't know one in which e is prime>
Math liesThis reminded me of a math blogosphere exchange recently where teachers were talking about the "little white lies in math." What do I mean:
- You can't subtract 7 from 3 (said when introducing subtraction and using a model of taking away physical objects.
- You can't divide 5 by 2 (distinguishing evens and odds, sharing whole objects, division within integers)
- You can't take a square root of -5
- You can't divide by 0
- You can't sum +1 - 1+1 -1 +1 -1 +....
- etc etc
I don't like it.
Ideally, we would take time to explore what it would mean if you could do those "forbidden" things. If time's not available now, offer to make time later. Maybe these are luxuries for someone who doesn't have to teach to an upcoming test, so if you can't explore with them, encourage the students to think about it on their own.
At the very least, use the right words to describe what you are (not) doing: "we can't take a square root of -5 in the real numbers." Sure, not all the students will get it, but hearing this caveat will clue them in that (a) this isn't a universal rule, so something special is happening, (b) there is more coming in the future and (c) what to do with their old understanding when they are finally shown the extension/clarification
Oh well, it seems to work for us.